Sufficient conditions for boundedness and continuity are obtained for stochastically continuous infinitely divisible processes, without Gaussian component, $ \{Y(t), t\in T\}$, where $T$ is a compact metric space or pseudo-metric space. Such processes have a version given by $ Y(t)=X(t)+b(t)$, $t\in T$ where $b$ is a deterministic drift function and $$ X(t)= \int_S f(t,s)\Big[N(ds)-(|f(t,s)|\vee 1)^{-1}\, \nu(ds)\Big]. $$ Here $N$ is a Poisson random measure on a Borel space $S$ with $\sigma$--finite mean measure $\nu$, and $f: T\times S\mapsto \R$ is a measurable deterministic function.
We show that the moving average process \[ {\mathcal X}_f(t):= \int_{0}^t f(t-s) \, dZ(s) \qquad t \in [0,T] \] has a bounded version almost surely, when the kernel $f$ has finite total 2--variation and $Z$ is a symmetric L\'evy process. We also obtain bounds for $E|\sup_{t\in[0,t]}{\mathcal X}_f(t)|$ and for uniform moduli of continuity of ${\mathcal X}_f(\cd)$ and for the largest jump of ${\mathcal X}_f(t)$ when it is not continuous. Similar results are obtained for forward average processes. The methods developed are also used to show that certain infinitely divisible random fields are bounded.
Given a positive measure $\mu$ on $R^n$, a continuous additive functional of an $R^n$ valued Levy process $X$ can be defined (heuristically) by \[ L^{\mu}_{t}=\lim_{\ep\to 0}\int \int_0^tf_\ep(X(s)-x)\,ds\,d\mu(x) \] where $f_\ep$ is an approximate $\de$ function at zero. If the support of $\mu$ is a curve in $R^2$ or a surface in $R^n$ we can consider $L_t^\mu$ as the local time on the curve or surface. Given a family $\MM$ of such measures, endowed with a metric, some sufficient conditions for the continuity of $\{L_t^\mu,(t,\mu)\in R_+\times\MM\}$ are obtained. This problem is related, in part, to the continuity of certain second order Gaussian chaos processes indexed by the functions in $\MM$.
Two results are given concerning the continuity and boundedness of stochastic convolutions of the form $$ {\mathcal X}_f(t) := \int_{0}^t f(t-s) \, dZ(s) \qquad t \in [0,1] $$ where $Z$ is a symmetric L\'evy process and $f: [0,1] \mapsto R$ is a continuous function with $f(0)=0$. The first is that the process ${\mathcal X}_f(t)$ is contiunous almost surely when $f$ is a sample path of any continuous Gaussian process with stationary increments, which is independent of $Z$. This shows, in particular, that there are continuous moving averages for which $f$ has arbitrarily large modulus of continuity. The second goes in the opposite direction. It shows that for any process $Z$, with paths of infinite variation, there exists a continuous function $f$, with $f(0)=0$, such that ${\mathcal X}_f(t)$ has unbounded paths almost surely.